December  2011, 31(4): 1219-1231. doi: 10.3934/dcds.2011.31.1219

Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem

1. 

SISSA, via Bonomea 265, 34136 Trieste, Italy

Received  December 2009 Revised  December 2010 Published  September 2011

We present a recent existence result concerning the quasistatic evolution of cracks in hyperelastic brittle materials, in the framework of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in [9].
Citation: Gianni Dal Maso, Giuliano Lazzaroni. Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1219-1231. doi: 10.3934/dcds.2011.31.1219
References:
[1]

L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in $SBV(\Omega,\R^k)$,, Nonlinear Anal., 23 (1994), 405.  doi: 10.1016/0362-546X(94)90180-5.  Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[3]

J. M. Ball, Some open problems in elasticity,, in, (2002), 3.   Google Scholar

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5.  doi: 10.1007/s10659-007-9107-3.  Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211.  doi: 10.1007/s00205-002-0240-7.  Google Scholar

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171.  doi: 10.1007/BF00250807.  Google Scholar

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[9]

G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257.   Google Scholar

[10]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results,, Arch. Ration. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

[11]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199.   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[13]

G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture,, Comm. Pure Appl. Math., 56 (2003), 1465.  doi: 10.1002/cpa.3039.  Google Scholar

[14]

G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem,, J. Mech. Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[15]

G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. Reine Angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[16]

N. Fusco, C. Leone, R. March and A. Verde, A lower semi-continuity result for polyconvex functionals in SBV,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 321.  doi: 10.1017/S0308210500004571.  Google Scholar

[17]

A. Giacomini and M. Ponsiglione, Non interpenetration of matter for $SBV$-deformations of hyperelastic brittle materials,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1019.  doi: 10.1017/S0308210507000121.  Google Scholar

[18]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163.   Google Scholar

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data,, Ann. Mat. Pura Appl. (4), 190 (2011), 165.  doi: 10.1007/s10231-010-0145-2.  Google Scholar

[20]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[21]

R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. Roy. Soc. London A, 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar

[22]

R. W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids,, Proc. Roy. Soc. London A, 328 (1972), 567.  doi: 10.1098/rspa.1972.0096.  Google Scholar

show all references

References:
[1]

L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in $SBV(\Omega,\R^k)$,, Nonlinear Anal., 23 (1994), 405.  doi: 10.1016/0362-546X(94)90180-5.  Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[3]

J. M. Ball, Some open problems in elasticity,, in, (2002), 3.   Google Scholar

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5.  doi: 10.1007/s10659-007-9107-3.  Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211.  doi: 10.1007/s00205-002-0240-7.  Google Scholar

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171.  doi: 10.1007/BF00250807.  Google Scholar

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[9]

G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257.   Google Scholar

[10]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results,, Arch. Ration. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

[11]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199.   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[13]

G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture,, Comm. Pure Appl. Math., 56 (2003), 1465.  doi: 10.1002/cpa.3039.  Google Scholar

[14]

G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem,, J. Mech. Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[15]

G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. Reine Angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[16]

N. Fusco, C. Leone, R. March and A. Verde, A lower semi-continuity result for polyconvex functionals in SBV,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 321.  doi: 10.1017/S0308210500004571.  Google Scholar

[17]

A. Giacomini and M. Ponsiglione, Non interpenetration of matter for $SBV$-deformations of hyperelastic brittle materials,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1019.  doi: 10.1017/S0308210507000121.  Google Scholar

[18]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163.   Google Scholar

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data,, Ann. Mat. Pura Appl. (4), 190 (2011), 165.  doi: 10.1007/s10231-010-0145-2.  Google Scholar

[20]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[21]

R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. Roy. Soc. London A, 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar

[22]

R. W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids,, Proc. Roy. Soc. London A, 328 (1972), 567.  doi: 10.1098/rspa.1972.0096.  Google Scholar

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